Continuum and Symmetry-Conserving Effects in Drip-line Nuclei Using Finite-range Forces

0.3 MB·English

by  N. Schunck

version of February 2, 2008
Continuum and Symmetry-Conserving Effects in Drip-line Nuclei Using Finite-range
Forces
N. Schunck1,2 and J. L. Egido1
1Departamento de Fisica Teorica, Universidad Autonoma de Madrid 28 049 Cantoblanco, Madrid, Spain
2University of Tennessee, Physics Division, 401 Nielsen Physics, Knoxville, TN 37996, USA
Oak Ridge National Laboratory, Bldg. 6025, MS6373, P.O. 6008, Oak Ridge, TN-37831, USA
(Dated: February 2, 2008)
We report the first calculations of nuclear properties near the drip-lines using the spherical
Hartree-Fock-Bogoliubov mean-field theory with a finite-range force supplemented by continuum
and particle numberprojection effects. Calculations were carried out in a basis made of the eigen-
states of a Woods-Saxon potential computed in a box, thereby garanteeing that continuum effects
wereproperlytakenintoaccount. Projectionoftheself-consistentsolutionsongoodparticlenumber
8
was carried out after variation, and an approximation of the variation after projection result was
0
used. We give the position of the drip-lines and examine neutron densities in neutron-rich nuclei.
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We discuss the sensitivity of nuclear observables upon continuum and particle-number restoration
2
effects.

n
a PACSnumbers: PACSnumbers: 21.10.Dr,21.10.Gv,21.10.Re,21.30.Fe,21.60.Ev,21.60.Jz
J

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2 One of the main challenges in contemporary nuclear ist when finite-range effective interactions are used [9].
structureistodescribenucleifarfromthevalleyofstabil- However, as of today, the description of weakly-bound

] ity. Newradioactiveionbeamfacilitiesareunderintense nuclei in the framework of finite-range mean-field the-
h
development and when in full exploitation will consider- orywas not available,as existing codes did not takeinto
t
- ably increase the wealth of experimental data available account continuum effects.
l
c [1]. From a theoretical point of view, neutron-rich or In this paper, we present the first attempt to extend
u
-deficient nuclei present two major challenges. Firstly, the mean-field approach based on finite-range interac-
n
these nuclei are weakly-bound, and it is expected that tions of the Gogny type to weakly-bound nuclei, by si-
[
theinfluenceofthecontinuumofenergywillsignificantly multaneously taking into account continuum effects and

2 altermanynuclearproperties[2]. However,theextentto the particle number symmetry. In section I we describe
v
which these properties will change due to the presence our method, which relies on the expansion of the mean-
0
of the continuum is not so clear. Secondly both the nu- field solutions on a realistic basis. In section II is shown
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clear mean-field theory and the nuclear shell model rely aselectionofresultsbenchmarkingourmethodandillus-
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0 on effective interactions whose parameters were mostly trating the influence of the continuum in weakly-bound
. adjusted on data in stable nuclei. The reliability and nuclei calculations. Projection on good particle number
0
the robustness of these parametrizations under extreme is discussed in section III.
1
isospin ratios must be questioned, and nuclei far from
7
0 stabilitythuspresentuswithauniquetest-ground[1,2].
: I. FINITE-RANGE MEAN-FIELD THEORIES
v Much effort was devoted in the recent years to the
WITH COUPLING TO THE CONTINUUM
i study of weakly-bound nuclei, and the development
X
of the corresponding appropriate methods, within the
r framework of the non-relativistic mean-field theory with Mean-field calculations with finite-range interactions
a
Skyrme interactions [3, 4, 5], relativistic mean-field [6] aremostconvenientlycarriedoutin configurationspace.
andthenuclearshellmodel[7]. Inmean-fieldapproaches, IntheHFBtheoryquasi-particlewave-functionsarethus
itwasrecognizedearlythatthestabilityofveryneutron- expanded on a basis of well-known functions. In nu-
rich nuclei is caused to a great extent by pairing corre- clear structure, the harmonic oscillator basis has always
lations, and that a fully self-consistent treatment of the playeda special role, as it produces localized eigenstates
latter in the Hartree-Fock-Bogoliubov formalism is re- thatareknownanalytically. However,awell-knowndefi-
quired[3]. InthecaseoftheSkyrmemean-fieldthezero- ciency of this basis is its improper asymptotic behavior.
range of the interaction leads to the well-known diver- Whilecontinuumstatesaresphericalwavesandtherefore
gence problem in the pairing channel, and practical cal- decay exponentially, HO functions behave like Gaussian
culationsmustresorttophenomenologicalregularization functions. Unless the number of basis functions is pro-
procedures[8]. Moreover,approachesbasedonthe effec- hibitively large, it is therefore impossible to describe all
tive density-functional theory allow terms to be present features of weakly-bound nuclei with such basis states.
in the functional, that do not originate from a two-body Apart from the techniques already mentioned [3, 4, 5,
Hamiltonian. The price to pay is the occurrence of di- 6], one simple and elegantwayto overcomethis problem
vergences when going beyond mean-field and restoring istousetheeigenstatesofa”realistic”nuclearpotential,
the particle number [9, 10]. These difficulties do not ex- that is, of a potential that tends to 0 at infinity. Such a
2
potential generates both a sequence of strongly localized the case of the most general potential, the existence of
bound-states and of unlocalized positive-energy states. such a simple criterion can not be guaranteed. Numer-
In practical calculations, boundary conditions must be ical tests showed that in heavy nuclei, ℓ = 15 and
max
set. Outgoingwaveboundaryconditionswithwavenum- n =18 allowed very accurate calculations from drip-
max
ber k assure that all solutions with positive energy are line to drip-line to within 10 keV. For the same required
taken into account (within the indispensable discretiza- accuracy, calculations performed with an energy cut-off
tion of the k values in practical numerical calculations), (all basis states with eigenvalues lower than E are re-
cut
including resonant and non-resonant states. However, tained) reached convergence for E of the order of 400
cut
the wave-functions are not square-integrable, and spe- MeV. About 92% of such a basis is made of positive-
cial techniques must be employed to introduce the com- energy unlocalized continuum states.
pleteness relation and calculate the matrix elements of The Gogny interactionis a finite-rangeforce that con-
the interaction on such a basis [11, 12, 13, 14]. On the tainacentralorBrink-Boekerterm,aspin-orbit,density-
otherhand,vanishingboundaryconditions,alternatively dependent and Coulomb terms. The spin-orbit and
named box boundary conditions, boil down to selecting Coulomb terms are the same as for zero-range Skyrme
onlythosepositive-energystatesthathaveanodeonthe forces [18]. Only a few parametrizations of this interac-
boundaries of the domain of integration. Consequently, tion exist. All of our calculations were carried out with
not the full continuum is taken into account, but the theD1Sforce[19]. Wewillstudy inaforthcomingpaper
wave-functions are orthonormal. For nuclear structure the behavior of the isovector part of the Gogny interac-
applications and the description of ground-state proper- tion in the region of the neutron-drip-line.
ties the box technique is sufficient and in fact equivalent
to the full continuum [15].
Our model potential was of the Woods-Saxon type,
which reads in spherical symmetry:
V
0
V(r)=− (1)
1+exp[(r−r )/a ]
0 0
The analytical solutions to the Schr¨odinger equation
with this potential are not known apart from s-waves
states [16]. However, they can be determined easily by
well-established numerical integration methods. We en-
close the potential in a sphere of radius R , discretize
box
the space with a mesh size h and integrate the radial
Schr¨odinger equation numerically:
HO (Analy.) HO (Box.) WS HO (Analy.) HO (Box.) WS
d2y 2m ℓ(ℓ+1)
nℓ
+ e − −V(r) y (r)=0 (2)
dr2 ~2 (cid:20) nℓ 2mr2 (cid:21) nℓ
FIG.1: Neutron(left)andproton(right)single-particleener-
Our tests showedthat a mesh size of h∼0.1 fm guaran- giesin208PbcalculatedintheHartree-Fockapproachwiththe
teed the convergence of the subsequent HFB calculation D1S Gogny effective interaction (ℓmax =15 and nmax =18).
to within a few keV. The first two columns in each figure correspond to the har-
The parameters of the potential do not play a major monic oscillator basis with N =24 shells, with either Gauss-
role here, as the eigenfunctions are only used as a basis. Hermite (’Analy.’) or numerical (’Box.’) integrations. The
Therefore, the particular form of the Woods-Saxon po- column on the right corresponds to theWoods-Saxon basis.
tential can only affect the speed at which convergence is
reachedbutnotthe finalHFBresult. We noticedthatin Figure 1 shows the single-particle spectrum in 208Pb
orderto obtaina fastconvergenceacrossthe whole mass around the Fermi level as function of the type of basis
table, the potential should contain a consequent amount used for the calculation. Benchmark calculations were
of bound-states. We therefore adopted the following performedintheHObasisusingstandardGauss-Hermite
parameters, that correspond to a realistic parametriza- quadrature formulas for the integrations (left column)
tion of the nuclear mean-field of element Z = 126 [17]: [20]. The middle column shows the results when the
V =−41.619 MeV, r =1.000 fm, a =1.693 fm. HO basis wave-functions are obtained by integrating the
0 0 0
Of more importance is the cut-off of the basis. Like in Schr¨odingerequationforaHOpotentialinaboxandus-
everynumericalimplementationofthemean-fieldtheory, ingnumericalintegrationtechniquestocomputethe ma-
the basis can not be infinite but must be limited artifi- trix elements of the Gogny interaction. Results obtained
cially. In practice, our basis functions are characterized in the WS basis are shown in the column on the right.
by two quantumnumbers, ℓ andn. Contraryto the har- As required, all three calculations give nearly identical
monic oscillator basis, where the main shell number N results for bound-states, within the numerical accuracy
provides a convenient and physical cut-off criterion, in of the integration.
3
Let us notice that, as the convergencein the HO basis II. DRIP-LINES AND CONTINUUM EFFECTS
depends on the number of shells and of the oscillator
length, the convergence in the WS basis depends on the
As an application of the consideration of continuum
cut-offs ℓ and n (or alternatively E ), on the
max max cut effects in practical calculations we present in Table I the
parameters of the WS potential itself and on the size of
neutronand protondrip-lines calculated in the spherical
the box. A detailed study of the convergence properties
HFBapproachfortheWoods-Saxonbasis. Forthelisted
of the WS basis will be presented elsewhere.
elements,thetwo-neutronsS andtwo-protonsS sep-
2n 2p
arationenergieswerecalculated,aswellasthepositionof
the Fermi level. The criterion to define the drip-line was
TABLE I: Table of Fermi level and separation energies in
drip-linenuclei either a positive separation energy, or a positive Fermi
level, whichever criterion was met first. The WS basis
Neutrons Protons
included all states with ℓ ≤ 15 and n ≤ 18 in a box of
λ −S λ −S
n 2n p 2p R =20 fm.
20C -1.39 2.92 10C -3.78 11.78 box
26O -0.36 1.16 14O -4.84 11.96 Drip-lines can be significantly pushed away by long-
30Ne -0.88 3.78 18Ne -2.38 5.18 range correlations. In particular, rare-earth nuclei are
40Mg -0.47 1.44 20Mg -1.09 2.66 foundtobestronglydeformedeitherinlarge-scaleGogny
46Si -0.43 1.50 24Si -2.06 6.24 HFB calculations using the D1S interaction [21] and in-
50S -0.17 1.30 28S -1.20 5.22 cluding beyond mean-field correlations [22] as well as in
56Ar -0.59 1.46 32Ar -1.46 4.18 Skyrme HFB calculations performed in the transformed
64Ca -0.09 0.06 36Ca -1.01 5.56 harmonic oscillator basis with the SLy4 parametrization
72Ti -0.18 0.66 40Ti -0.27 1.78 in the mean-field channel and the volume delta pairing
76Cr -0.16 0.62 44Cr -1.21 3.24 force [23]. Bearingthis fact in mind, we nevertheless ob-
82Fe -0.26 0.86 46Fe -0.17 1.02 servesystematicdifferencesonthepositionoftheneutron
86Ni -0.22 0.98 52Ni -1.18 5.41 drip-line as compared to Skyrme HFB calculations pub-
92Zn -0.18 0.60 58Zn -0.70 2.53 lished in [23] where the drip-line almost systematically
104Ge -0.12 0.14 62Ge -0.77 3.22
extends further away by 2-4 neutrons.
114Se -0.33 0.76 66Se -0.89 3.54
116Kr -1.10 2.24 68Kr -0.08 0.71
120Sr -0.44 2.76 72Sr -0.54 1.88
122Zr -1.02 4.36 78Zr -1.12 3.37 0
130Mo -0.02 0.16 82Mo -0.38 2.90 10
136Ru -0.01 0.14 86Ru -0.63 2.12
-2
140Pd -0.08 0.42 90Pd -0.86 2.45 10
152Cd -0.03 0.04 94Cd -1.05 2.66 3]
111118787708406BXCSTneaee -----00001.....4209546391 03013.....9119006282 111111120184220BXCSTneaee -----00000.....0571340097 11611.....5804630632 -[utron Density fm 111000---864 111135732002SSSnnn --- HHHOOO
186Nd -0.65 4.10 126Nd -0.52 1.79 Ne Sn - HO
188Sm -1.08 5.06 130Sm -0.41 1.59 150Sn - HO
190Gd -1.52 6.00 134Gd -0.04 1.32 10-10 172Sn - HO
198Dy -0.10 0.38 140Dy -0.34 1.39 150Sn - Ref
206Er -0.02 0.04 144Er -0.19 1.14 10-12
220Yb -0.02 0.02 148Yb -0.11 0.90 0 5 10 15 20
240Hf -0.03 0.12 152Hf -0.05 0.72 Radius r [fm]
252W -0.06 0.20 158W -0.59 1.76
258Os -0.15 0.32 162Os -0.49 1.48 FIG. 2: (color online) Neutron densities in neutron-rich
260Pt -0.44 0.88 166Pt -0.38 1.17 132,150,170Sn isotopes computed with the finite range D1S
262Hg -0.71 1.42 170Hg -0.25 0.79 Gogny interaction in the HO and Woods-Saxon bases for a
264Pb -1.00 1.46 182Pb -0.08 4.12 box of 20 fm. Thecurvemarked Ref. is theHFBcalculation
268Po -0.00 1.70 194Po -0.16 1.84 incoordinate spacefor 150SnwiththeSkPforce [3]. Theun-
270Rn -0.24 2.22 198Rn -0.07 1.11 physicalasymptoticbehaviorofdensitiescomputedintheHO
272Ra -0.50 2.72 204Ra -0.43 1.70 basis is clearly visible. Oscillations in the case of 132Sn are
274Th -0.76 3.22 208Th -0.21 1.26 caused by the absence of pairing correlations which forbids
276U -1.04 3.76 214U -0.32 1.56 ℓ-couplings in theexpansion.
282Pu -0.06 0.28 220Pu -0.42 1.54
296Cm -0.04 0.04 222Cm -0.17 1.11
Observables like densities are very sensitive to contin-
uum effects and significantly impact many spectroscopic
or reaction properties of nuclei. A stringent test of our
4
methodisthereforeitsabilityto reproducethe longtails |φ(x)i corresponding to the self-consistent diagonaliza-
of neutron densities in neutron-rich nuclei. There have tion of the one-body Hamiltonians hˆ (x), as a func-
HFB
beennumerousstudiesofneutronskinsinexoticnucleiin tion of x. Here hˆ (x) stands for the HFB Hamil-
HFB
the past, andwe cantherefore benchmark ourapproach.
tonian obtained by multiplying, at each iteration, the
Asanexamplewedisplayinfigure2,theneutronden- pairing field ∆ of the HFB matrix by the constant x,
sities of three very neutron-rich Sn isotopes. For small i.e., ∆ → ∆(x) = x∆, while the particle-hole field h
andmediumvaluesofthenuclearradiusthe HOandthe remains unchanged. To each wave function |φ(x)i is as-
WS bases give the same density profile. However, for sociated an eigenstate of the particle number operator
large radius the differences are significant. As expected, |Φ(x)i=PˆN|φ(x)i and an energy
the proper asymptotic of the WS basis wave-functions is
reflected in the long density tails. It is important to em- hΦ(x)|Hˆ|Φ(x)i
EN(x)= (4)
phasizethattheresultsarenearlyidenticaltocoordinate
hΦ(x)|Φ(x)i
spaceSkyrmeHFBcalculationsof[3]ascanbeseenfrom
the curve markedRef., which shows the density in 150Sn with Hˆ the original two-body Hamiltonian. The varia-
obtainedwiththeSkPparametrizationoftheSkyrmein- tionalprincipleguaranteesthatsuchaprocedureyieldsa
teraction. This provides an independent validation and minimum as function of x, which is an excellent approx-
veryrobusttestofourmethodtoincorporatecontinuum imation to the VAP result [25]. We call this approach
effects through a basis. RVAP (Restricted VAP).
Weobserveoscillationsinthedensitiesforclosed-shells
nuclei like 132Sn. Indeed, in that case the density is sim-
plythesumofN well-localizedsingle-particlestates,and
no contributionfromdiscretizedcontinuumstatesenters 42
Mg - Spherical RVAP
theexpansion. Thethresholdwheretheseoscillationsap- -256.5
peargivesinfactameasureoftheintrinsicnumericalac-
V]
curacywith whichintegrationsareperformed. This phe- e
M
nomenon may be distinguished from what was reported y [-257.0
in [24] where oscillations were caused by large discrep- erg
ancies between the upper and lower components in the En
HFB equations. d -257.5
e
ct
e
oj
Pr-258.0
III. RESTORATION OF THE PARTICLE
NUMBER SYMMETRY
-258.5
0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35
As mentioned in the introduction pairing correlations
play a relevant role in the description of weakly bound Pairing Enhancement Factor x
nuclei. Uptonowmostdescriptionsofthedrip-linewere
performed in the HFB approach which has two draw- FIG.3: (color online) Total projected energyEN(x)in42Mg
backs. Firstly, in the weak pairing regime it usually col- as function of the effective pairing strength factor x. The
lapses to the unpaired solution and secondly, the HFB RVAPcalculation was performed in theWS basis.
wave function has the right number of particles only on
the average. To restore this spontaneously broken sym- Toinvestigatetheimpactofabettertreatmentofpair-
metry, we project the HFB solution onto good particle ingcorrelationsonexoticnucleiweappliedthisrestricted
number using the projector: VAPprocedureinthe neighborhoodofthe neutrondrip-
line. Figure 3 shows the example of 42Mg which is un-
PˆN = 1 2πdϕeiϕ(Nˆ−N) (3) bound against2 neutrons emission in standardspherical
2π Z HFB or PAV-HFB calculations. The minimum of the
0
curve depicted in Fig. 3 corresponds to a stronger pair-
according to the procedure described in details in [9]. ing regime than in standard HFB calculations: the total
As is well-known, the variation after projection (VAP) gaininenergyisonly460keVbutthepairingenergygoes
method provides a much better approach than the pro- from-9.22MeVto-13.83MeVfromx=1.0tox=1.12.
jectionaftervariation(PAV)atthecostofamuchhigher Atthe minimum, the two-neutronseparationenergycal-
numerical effort, especially with the huge basis used in culated from the RVAP projected energies of 42Mg and
the present calculations. In particular, it prevents pair- 40Mg becomes negative at S = −252 keV (compared
2n
ing correlations to collapse. Nevertheless one can obtain with S = +40 keV in the un-projected case) and the
2n
an approximate VAP solution by searching for the min- nucleus is bound against 2 neutrons emission. This ef-
imum in a restricted highly pair-correlated variational fectisusuallytypicalfromnucleinearclosed-shellswhere
space [25]. This space is generated by HFB solutions onlytheVAPmethodcanpreventtheHFBtheorytocol-
5
lapse to the unpaired regime. Near the drip-lines, it also we find that the asymptotic of the neutron densities is
results in a visible shift of the drip-lines. It is impor- properly reproduced. The role of the restoration of par-
tant to realize that this effect can not appear in a PAV ticle number before variation is emphasized. It is found
calculation. that this mechanism can stabilize nuclei at the drip-line
Therestorationofsymmetriesbrokenatthemean-field thereby pushing away the latter.
level has therefore a significant and experimentally vis-
ible effect in exotic neutron-rich nuclei. Variation after Acknowledgment- WewishtothankP.RingandT.
projectionon goodparticle number consists in exploring R.Rodriguezforfruitfuldiscussions. Oneofus(N.S.)ac-
a different variational space than when the projection is knowledgesfinancialsupportofthespanishMinisteriode
carried out after the variation. Educaciony Ciencia (Ref. SB2004-0024). This workhas
In summary, we present the first results of a general- been supported in part by the DGI, Ministerio de Cien-
izationofHFBcalculationswithfinite-rangeinteractions ciayTecnologia,Spain,underProjectFIS2004-06697,as
to exotic weakly-bound nuclei. Mean-field solutions are wellasbytheU.S.DepartmentofEnergyunderContract
expandedonabasisofthe eigenstatesofaWoods-Saxon Nos. DE-FC02-07ER41457 (University of Washington),
potential,whichpossesstheproperasymptoticbehavior. DEFG02-96ER40963(UniversityofTennessee),andDE-
In stable nuclei, this WS basis is benchmarked against AC05- 00OR22725 with UT-Battelle, LLC (Oak Ridge
standard harmonic oscillator calculations and is found National Laboratory). We thank K. Bennaceur and J.
to give exactly the same results. In neutron-rich nuclei Dobaczewski for letting us use the code HFBRAD.
[1] J. Dobaczewski and W. Nazarewicz Phil. Trans. R. Soc. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965); W. J.
Lond.A 356, 2007-2031 (1998) Romo, Nucl.Phys. A116, 617 (1968);
[2] J. Dobaczewski, N. Michel, W. Nazarewicz, M. [12] T. Berggren, Nucl. Phys.A109, 265 (1968);
Pl oszajczak andJ.RotureauProg.Part.Nucl.Phys.56, [13] B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523
432-455 (2007) (1971); G. Garcia-Calderon and R. Peierls, Nucl. Phys.
[3] J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys. A265, 443 (1976);
A422, 103 (1984); J. Dobaczewski, W. Nazarewicz, T. [14] P. Lind, Phys.Rev. C471903 (1993)
R. Werner, J.-F. Berger, C. R. Chinn and J. Decharg´e, [15] M. Stoitsov, N. Michel and K. Matsuyanagi, nucl-
Phys.Rev.C53, 2809-2840 (1996) th/arXiv:0709.1006
[4] M.V.Stoitsov,W.NazarewiczandS.Pittel,Phys.Rev. [16] S. Flu¨gge, Practical Quantum Mechanics I (Springer-
C58, 2092 (1998); M. V. Stoitsov, J. Dobaczewski, W. Verlag, Berlin, 1971) vol. 1, Prob. No. 37, 39.
Nazarewicz and P. Ring, Comput. Phys. Commun. 167, [17] J.DudekandT.R.Werner,Phys.Lett.10,1543 (1978)
43-63 (2005) [18] J.Decharg´eandD.Gogny,Phys.Rev.C21,1568(1980)
[5] M.Grasso, N.Sandulescu,N.VanGiaiandR.J.Liotta, [19] J.-F. Berger, M. Girod and D. Gogny, Comput. Phys.
Phys.Rev.C64, 064321 (2001) Commun. 63, 365 (1991)
[6] W. Poeschl, D. Vretenar and P. Ring, Comp. Phys. [20] T. Gonzalez-Llarena, Ph.D Thesis, Universidad Au-
Comm. 103, 217 (1997) tonoma deMadrid, Sep.1999, unpublished.
[7] N. Michel, W. Nazarewicz, M. Pl oszajczak and J. [21] http://www-phynu.cea.fr/science en ligne/carte poten-
Okol owicz Phys. Rev. C67, 054311 (2003); N. Michel, tiels microscopiques/carte potentiel nucleaire.htm
W.Nazarewicz, M.Pl oszajczak andK.BennaceurPhys. [22] T. R. Rodriguez and J. L. Egido, Phys. Rev. Lett. 99,
Rev.Lett. 89, 042502 (2002) 062501 (2007)
[8] P. Borycki, J. Dobaczewski, W. Nazarewicz and M. [23] M.V.Stoitsov,J.Dobaczewski,W.Nazarewicz,S.Pittel
Stoitsov Phys. Rev.C73, 044319 (2006) and D.J. Dean, Phys.Rev. C68, 054312 (2003)
[9] M. Anguiano, J.L. Egido and L.M. Robledo Nucl.Phys. [24] K.BennaceurandJ.Dobaczewski, Comp. Phys.Comm.
A696, 467-493 (2001); M. Anguiamo, J.L. Egido and 168, 96-122 (2005)
L.M. Robledo Phys. Lett.B545, 62-72 (2002); [25] T. R. Rodriguez, J. L. Egido and L. M. Robledo, Phys.
[10] J. Dobaczewski, M.V. Stoitsov, W. Nazarewicz, P.-G. Rev. C72, 064303 (2005)
Reinhard,nucl-th/arXiv:0708.0441
[11] Y.B. Zel’dovich, Sov. Phys. JETP. 12, 542 (1960); N.